Shelton, Brad
Kloefkorn, Tyler
2014-09-29T17:46:46Z
2014-09-29
http://hdl.handle.net/1794/18372
This dissertation studies new connections between combinatorial topology and homological algebra. To a finite ranked poset Γ we associate a finite-dimensional quadratic graded algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the splitting algebra AΓ. First introduced by Gelfand, Retakh, Serconek and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials.
Given a finite ranked poset Γ, we ask a standard question in homological algebra: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ known as Cohen-Macaulay. One of the main theorems of this dissertation is: If Γ is a finite ranked cyclic poset, then Γ is Cohen-Macaulay if and only if Γ is uniform and RΓ is Koszul.
We also define a new generalization of Cohen-Macaulay: weakly Cohen-Macaulay. The class of weakly Cohen-Macaulay finite ranked posets includes posets with disconnected open subintervals. We prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul.
Finally, we address the notion of numerical Koszulity. We show that there exist algebras RΓ that are numerically Koszul but not Koszul and give a general construction for such examples.
This dissertation includes unpublished co-authored material.
en_US
University of Oregon
All Rights Reserved.
Cohen-Macaulay
Koszul
Splitting Algebras
On Algebras Associated to Finite Ranked Posets and Combinatorial Topology: The Koszul, Numerically Koszul and Cohen-Macaulay Properties
Electronic Thesis or Dissertation
2015-03-29
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Giusti, Chad David, 1978-
2010-12-03T22:07:48Z
2010-12-03T22:07:48Z
2010-06
http://hdl.handle.net/1794/10869
viii, 57 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We introduce a new finite-complexity knot theory, the theory of plumbers' knots, as a model for classical knot theory. The spaces of plumbers' curves admit a combinatorial cell structure, which we exploit to algorithmically solve the classification problem for plumbers' knots of a fixed complexity. We describe cellular subdivision maps on the spaces of plumbers' curves which consistently make the spaces of plumbers' knots and their discriminants into directed systems.
In this context, we revisit the construction of the Vassiliev spectral sequence. We construct homotopical resolutions of the discriminants of the spaces of plumbers knots and describe how their cell structures lift to these resolutions. Next, we introduce an inverse system of unstable Vassiliev spectral sequences whose limit includes, on its E ∞ - page, the classical finite-type invariants. Finally, we extend the definition of the Vassiliev derivative to all singularity types of plumbers' curves and use it to construct canonical chain representatives of the resolution of the Alexander dual for any invariant of plumbers' knots.
Committee in charge: Dev Sinha, Chairperson, Mathematics;
Hal Sadofsky, Member, Mathematics;
Arkady Berenstein, Member, Mathematics;
Daniel Dugger, Member, Mathematics;
Andrzej Proskurowski, Outside Member, Computer & Information Science
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2010;
Plumbers' knots
Vassiliev derivatives
Finite-complexity knots
Spectral sequences
Alexander dual
Canonical chains
Mathematics
Theoretical mathematics
Plumbers' knots and unstable Vassiliev theory
Thesis

Brandl, Mary-Katherine, 1963-
2008-02-10T04:19:31Z
2008-02-10T04:19:31Z
2001
0-493-36423-4
http://hdl.handle.net/1794/147
Adviser: Brad Shelton.
viii, 49 leaves
A print copy of this title is available through the UO Libraries under the call number: MATH LIB. QA251.3 .B716 2001
We examine a family of twists of the complex polynomial ring on n generators by a non-semisimple automorphism. In particular, we consider the case where the automorphism is represented by a single Jordan block. The multiplication in the twist determines a Poisson structure on affine n-space. We demonstrate that the primitive ideals in the twist are parameterized by the symplectic leaves associated to this Poisson structure. Moreover, the symplectic leaves are determined by the orbits of a regular unipotent subgroup of the complex general linear group.
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en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2001
Polynomial rings
Poisson algebras
Noncommutative rings
Primitive and Poisson spectra of non-semisimple twists of polynomial algebras
Thesis

Ahlquist, Blair, 1979-
2011-05-04T01:19:26Z
2011-05-04T01:19:26Z
2010-09
http://hdl.handle.net/1794/11144
vi, 48 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We compare the relaxation times of two random walks - the simple random walk and the metropolis walk - on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is included with probability p = [Special characters omitted.] where λ > 1 is a constant and also to the Newman-Watts small world model. We give a bound for the reconstruction problem for general trees and general 2 × 2 matrices in terms of the branching number of the tree and some function of the matrix. Specifically, if the transition probabilities between the two states in the state space are a and b , we show that we do not have reconstruction if Br( T ) [straight theta] < 1, where [Special characters omitted.] and Br( T ) is the branching number of the tree in question. This bound agrees with a result obtained by Martin for regular trees and is obtained by more elementary methods. We prove an inequality closely related to this problem.
Committee in charge: David Levin, Chairperson, Mathematics;
Christopher Sinclair, Member, Mathematics;
Marcin Bownik, Member, Mathematics;
Hao Wang, Member, Mathematics;
Van Kolpin, Outside Member, Economics
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2010;
Probability
Graphs
Random walks
Reconstruction problem
Metropolis walk
Mixing time
Probability on graphs: A comparison of sampling via random walks and a result for the reconstruction problem
Thesis

Berenstein, Arkady
Rupel, Dylan
Rupel, Dylan
2012-10-26T04:01:34Z
2012-10-26T04:01:34Z
2012
http://hdl.handle.net/1794/12400
We de ne the quantum cluster character assigning an element of a quantum torus to each
representation of a valued quiver (Q; d) and investigate its relationship to external and internal
mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations
are related to re
ection functors and internal mutations are related to tilting theory. Our
main result will show the quantum cluster character gives a cluster monomial in this quantum
cluster algebra whenever the representation is rigid, moreover we will see that each non-initial
cluster variable can be obtained in this way from the quantum cluster character.
en_US
University of Oregon
All Rights Reserved.
Cluster
Quantum
Quiver
Tilting
Quantum Cluster Characters
Electronic Thesis or Dissertation

Black, Samson, 1979-
2010-11-30T01:26:26Z
2010-11-30T01:26:26Z
2010-06
http://hdl.handle.net/1794/10847
viii, 50 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We study a certain quotient of the Iwahori-Hecke algebra of the symmetric group Sd , called the super Temperley-Lieb algebra STLd. The Alexander polynomial of a braid can be computed via a certain specialization of the Markov trace which descends to STLd. Combining this point of view with Ocneanu's formula for the Markov trace and Young's seminormal form, we deduce a new state-sum formula for the Alexander polynomial. We also give a direct combinatorial proof of this result.
Committee in charge: Arkady Vaintrob, Co-Chairperson, Mathematics
Jonathan Brundan, Co-Chairperson, Mathematics;
Victor Ostrik, Member, Mathematics;
Dev Sinha, Member, Mathematics;
Paul van Donkelaar, Outside Member, Human Physiology
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2010;
Hecke algebras
Alexander polynomal
Symmetric groups
Markov trace
Mathematics
Theoretical mathematics
Representations of Hecke algebras and the Alexander polynomial
Thesis

Kleshchev, Alexander
Muth, Robert
2016-10-27T18:33:59Z
2016-10-27T18:33:59Z
2016-10-27
http://hdl.handle.net/1794/20432
We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.
en_US
University of Oregon
All Rights Reserved.
KLR algebras
Representation theory
Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Brundan, Jon
Reynolds, Andrew
2015-08-18T22:59:49Z
2015-08-18T22:59:49Z
2015-08-18
http://hdl.handle.net/1794/19228
We study the representations of a certain specialization $\mathcal{OB}(\delta)$ of the oriented Brauer category in arbitrary characteristic $p$. We exhibit a triangular decomposition of $\mathcal{OB}(\delta)$, which we use to show its irreducible representations are labelled by the set of all $p$-regular bipartitions. We then explain how its locally finite dimensional representations can be used to categorify the tensor product $V(-\varpi_{m'}) \otimes V(\varpi_{m})$ of an integrable lowest weight and highest weight representation of the Lie algebra $\mathfrak{sl}_{\Bbbk}$. This is an example of a slight generalization of the notion of tensor product categorification in the sense of Losev and Webster and is the main result of this paper. We combine this result with the work of Davidson to describe the crystal structure on the set of irreducible representations. We use the crystal to compute the decomposition numbers of standard modules as well as the characters of simple modules assuming $p = 0$. We give another proof of the classification of irreducible modules over the walled Brauer algebra. We use this classification to prove that the irreducible $\mathcal{OB}(\delta)$-modules are infinite dimensional unless $\delta = 0$, in which case they are all infinite dimensional except for the irreducible module labelled by the empty bipartition, which is one dimensional.
en_US
University of Oregon
All Rights Reserved.
Representations of the Oriented Brauer Category
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Kronholm, William C., 1980-
2009-01-13T00:36:10Z
2009-01-13T00:36:10Z
2008-06
http://hdl.handle.net/1794/8284
x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )-graded equivariant cohomology of G -spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces.
In addition, the cohomology of Rep( G )-complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes.
Adviser: Daniel Dugger
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2008;
Algebraic topology
Equivariant topology
Spectral sequence
Serre spectral sequence
Mathematics
The RO(G)-graded Serre Spectral Sequence
Thesis

Berenstein, Arkady
Foster, John
2013-10-03T23:33:29Z
2013-10-03T23:33:29Z
2013-10-03
http://hdl.handle.net/1794/13269
We exhibit a correspondence between subcategories of modules over an algebra and sub-bimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence of a Peter-Weyl decomposition of the corresponding sub-bimodule. Finally, we use this technique to establish the semisimplicity of certain finite-dimensional representations of the quantum double $D(U_q(sl_2))$ for generic $q$.
en_US
University of Oregon
All Rights Reserved.
Semisimplicity of Certain Representation Categories
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Sinclair, Christopher
Shum, Christopher
2013-10-03T23:35:27Z
2013-10-03T23:35:27Z
2013-10-03
http://hdl.handle.net/1794/13302
For beta > 0, the beta-ensemble corresponds to the joint probability density on the real line proportional to
prod_{n > m}^N abs{x_n - x_m}^beta prod_{n = 1}^N w(x_n)
where w is the weight of the system. It has the application of being the Boltzmann factor for the configuration of N charge-one particles interacting logarithmically on an infinite wire inside an external field Q = -log w at inverse temperature beta. Similarly, the circular beta-ensemble has joint probability density proportional to
prod_{n > m}^N abs{e^{itheta_n} - e^{itheta_m}}^beta prod_{n = 1}^N w(x_n) quad for theta_n in [- pi, pi)
and can be interpreted as N charge-one particles on the unit circle interacting logarithmically with no external field. When beta = 1, 2, and 4, both ensembles are said to be solvable in that their correlation functions can be expressed in a form which allows for asymptotic calculations. It is not known, however, whether the general beta-ensemble is solvable.
We present four families of particle models which are solvable point processes related to the beta-ensemble. Two of the examples interpolate between the circular beta-ensembles for beta = 1, 2, and 4. These give alternate ways of connecting the classical beta-ensembles besides simply changing the values of beta. The other two examples are "mirrored" particle models, where each particle has a paired particle reflected about some point or axis of symmetry.
en_US
University of Oregon
All Rights Reserved.
Beta Ensemble
Random Matrix Theory
Solvable Particle Models Related to the Beta-Ensemble
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Leeman, Aaron, 1974-
2010-03-01T23:23:03Z
2010-03-01T23:23:03Z
2009-06
http://hdl.handle.net/1794/10227
vii, 34 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We study the Bousfield localization functors known as [Special characters omitted], as described in [MahS]. In particular we would like to understand how they interact with suspension and how they stabilize.
We prove that suitably connected [Special characters omitted]-acyclic spaces have suspensions which are built out of a particular type n space, which is an unstable analog of the fact that [Special characters omitted]-acyclic spectra are built out of a particular type n spectrum. This theorem follows Dror-Farjoun's proof in the case n = 1 with suitable alterations. We also show that [Special characters omitted] applied to a space stabilizes in a suitable way to [Special characters omitted] applied to the corresponding suspension spectrum.
Committee in charge: Hal Sadofsky, Chairperson, Mathematics;
Arkady Berenstein, Member, Mathematics;
Daniel Dugger, Member, Mathematics;
Dev Sinha, Member, Mathematics;
William Rossi, Outside Member, English
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2009;
Chromatic functors
Bousfield functors
Acyclic spaces
Suspension spectrum
Algebraic topology
Mathematics
Stabilization of chromatic functors
Thesis

Jordan, Alex, 1979-
2009-01-13T00:17:10Z
2009-01-13T00:17:10Z
2008-06
http://hdl.handle.net/1794/8283
vii, 41 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We generalize a theorem of Zhu relating the trace of certain vertex algebra representations and modular invariants to the arena of vertex super algebras. The theorem explains why the space of simple characters for the Neveu-Schwarz minimal models NS( p, q ) is modular invariant. It also expresses negative products in terms of positive products, which are easier to compute. As a consequence of the main theorem, the subleading coefficient of the singular vectors of NS( p, q ) is determined for p and q odd. An interesting family of q -series identities is established. These consequences established here generalize results of Milas in this field.
Adviser: Arkady Vaintrob
en_US
University of Oregon
University of Oregon theses, Dept. of Mathematics, Ph. D., 2008;
Vertex algebras
Neveu-Schwarz model
Super algebras
Zhu's theorem
Mathematics
A Super Version of Zhu's Theorem
Thesis

Isenberg, James
Dilts, James
2015-08-18T23:00:52Z
2015-08-18T23:00:52Z
2015-08-18
http://hdl.handle.net/1794/19237
In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures.
This dissertation includes previously published coauthored material.
en_US
University of Oregon
All Rights Reserved.
differential geometry
general relativity
partial differential equations
The Einstein Constraint Equations on Asymptotically Euclidean Manifolds
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

He, Weiyong
Welly, Adam
2016-10-27T18:40:22Z
2016-10-27T18:40:22Z
2016-10-27
http://hdl.handle.net/1794/20466
Let (M,g) be a quasi-Sasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain non-negativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g).
Naturally associated to a quasi-Sasaki metric g is a transverse Kahler metric g^T. The transverse Kahler-Ricci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in Kahler-Ricci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting.
We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasi-Sasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is eta-Einstein.
en_US
University of Oregon
All Rights Reserved.
differential geometry
Einstein metric
Kahler
quasi-Sasaki
Ricci flow
Sasaki
The Geometry of quasi-Sasaki Manifolds
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Sadofsky, Hal
Vicinsky, Deborah
2015-08-18T23:06:22Z
2015-08-18T23:06:22Z
2015-08-18
http://hdl.handle.net/1794/19283
We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the Bisson-Tsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories.
en_US
University of Oregon
All Rights Reserved.
Algebraic topology
Goodwillie calculus
Homotopy theory
Model categories
The Homotopy Calculus of Categories and Graphs
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Levin, David
Montgomery, Aaron
2013-10-03T23:37:50Z
2013-10-03T23:37:50Z
2013-10-03
http://hdl.handle.net/1794/13335
We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.
en_US
University of Oregon
All Rights Reserved.
balanced incomplete block designs
collisions of random walks
Markov chains
Topics in Random Walks
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Lin, Huaxin
Sun, Michael
2014-09-29T17:46:18Z
2014-09-29T17:46:18Z
2014-09-29
http://hdl.handle.net/1794/18368
In this dissertation we explore the question of existence of a property of group actions on C*-algebras known as the tracial Rokhlin property. We prove existence of the property in a very general setting as well as specialise the question to specific situations of interest.
For every countable discrete elementary amenable group G, we show that there always exists a G-action ω with the tracial Rokhlin property on any unital simple nuclear tracially approximately divisible C*-algebra A. For the ω we construct, we show that if A is unital simple and Z-stable with rational tracial rank at most one and G belongs to the class of countable discrete groups generated by finite and abelian groups under increasing unions and subgroups, then the crossed product A 􏰃ω G is also unital simple and Z-stable with rational tracial rank at most one.
We also specialise the question to UHF algebras. We show that for any countable discrete maximally almost periodic group G and any UHF algebra A, there exists a strongly outer product type action α of G on A. We also show the existence of countable discrete almost abelian group actions with the "pointwise" Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear C*-algebras with tracial rank zero and a unique tracial state appearing as crossed products.
en_US
University of Oregon
All Rights Reserved.
C*-algebras
classification
crossed product
existence
group actions
tracial Rokhlin property
The Tracial Rokhlin Property for Countable Discrete Amenable Group Actions on Nuclear Tracially Approximately Divisible C*-Algebras
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon

Lu, Peng
Bell, Thomas
2013-10-03T23:31:15Z
2013-10-03T23:31:15Z
2013-10-03
http://hdl.handle.net/1794/13231
In the first chapter we consider the question of uniqueness of conformal Ricci flow. We use an energy functional associated with this flow along closed manifolds with a metric of constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of the solution to both the metric and the pressure function along conformal Ricci flow.
In the next chapter we study backward Ricci flow of locally homogeneous
geometries of 4-manifolds which admit compact quotients. We describe the longterm behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.
en_US
University of Oregon
All Rights Reserved.
Conformal
Differential
Flow
Geometry
Homogeneous
Ricci
Uniqueness of Conformal Ricci Flow and Backward Ricci Flow on Homogeneous 4-Manifolds
Electronic Thesis or Dissertation
Ph.D.
doctoral
Department of Mathematics
University of Oregon